Beta distribution

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Beta
Probability density function Image:Beta distribution pdf.png
Cumulative distribution function Image:Beta distribution cdf.png
Parameters	<math>\alpha > 0</math> shape (real) <math>\beta > 0</math> shape (real)
Support	<math>x \in [0; 1]\!</math>
Probability density function (pdf)	<math>\frac{x^{\alpha-1}(1-x)^{\beta-1
Cumulative distribution function (cdf)	{{{cdf}}}
Mean	{{{mean}}}
Median	{{{median}}}
Mode	{{{mode}}}
Variance	{{{variance}}}
Skewness	{{{skewness}}}
Excess Kurtosis	{{{kurtosis}}}
Entropy	{{{entropy}}}
mgf	{{{mgf}}}
Char. func.	{{{char}}}

{\mathrm{B}(\alpha,\beta)}\!</math>|

 cdf        =<math>I_x(\alpha,\beta)\!</math>|
 mean       =<math>\frac{\alpha}{\alpha+\beta}\!</math>|
 median     =|
 mode       =<math>\frac{\alpha-1}{\alpha+\beta-2}\!</math> for <math>\alpha>1, \beta>1</math>|
 variance   =<math>\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\!</math>|
 skewness   =<math>\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}</math>|
 kurtosis   =see text|
 entropy    =<math>\ln\mathrm{\Beta}(a,b)-(a-1)\psi(a)-(b-1)\psi(b)+(a+b-2)\psi(a+b)</math>|
 mgf        =<math>1  +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}</math>|
 char       =<math>{}_1F_1(\alpha; \alpha+\beta; i\,t)\!</math>|

}} In probability theory and statistics, the beta distribution is a two-parameter family of continuous probability distributions defined on the interval [0, 1], with probability density function (pdf)

<math> f(x;\alpha,\beta) = \frac{1}{\mathrm{B}(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}. \,\!</math>

Parameters α and β must be greater than zero, and B is the beta function.

1 Properties
2 Shapes
3 Related distributions
4 Applications
5 External links

[edit] Properties

[edit] Normalization

The beta function appears as a normalization constant simply to ensure that the integral of the pdf is unity:

<math> f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int_0^1 u^{\alpha-1} (1-u)^{\beta-1}\, du} \!</math>

<math>= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>

<math>= \frac{1}{\mathrm{B}(\alpha,\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!</math>

where Γ is the gamma function.

[edit] Moments

The expected value and variance of a beta random variable X with parameters α and β are given by the formulae:

<math> \operatorname{E}(X) = \frac{\alpha}{\alpha+\beta} </math>

<math> \operatorname{Var}(X) = \frac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}</math>

This relationship can be inverted. When the expected value and variance of a beta random variable X are given, the parameters α and β are calculated by the formulae:

<math> \alpha = \operatorname{E}(X)

\left(

\frac{\operatorname{E}(X) (1 - \operatorname{E}(X))}{\operatorname{Var}(X)} - 1

\right), \,\!</math>

<math>

\beta = (1-\operatorname{E}(X)) \left(

\frac{\operatorname{E}(X) (1 - \operatorname{E}(X))}{\operatorname{Var}(X)} - 1

\right). \,\!</math>

If the sample mean and sample variance are put in place of E(X) and Var(X), then the result values of α and β are estimates of those parameters by the method of moments.

For any two numbers u, v such that 0 < u < 1 and 0 < v < u(1 − u) there is a beta distribution having expected value E(X) = u and variance Var(X) = v.

The skewness is

<math>

\frac{2 (\beta - \alpha) \sqrt{\alpha + \beta + 1} }

       {(\alpha + \beta + 2) \sqrt{\alpha \beta}}. \,\!

</math>

The kurtosis excess is:

<math>6\,\frac{\alpha^3-\alpha^2(2\beta-1)+\beta^2(\beta+1)-2\alpha\beta(\beta+2)}

{\alpha \beta (\alpha+\beta+2) (\alpha+\beta+3)}.\,\!</math>

[edit] Cumulative distribution function

The cumulative distribution function is

<math>F(x;\alpha,\beta) = \frac{\mathrm{B}_x(\alpha,\beta)}{\mathrm{B}(\alpha,\beta)} = I_x(\alpha,\beta) \!</math>

where <math>\mathrm{B}_x(\alpha,\beta)</math> is the incomplete beta function and <math>I_x(\alpha,\beta)</math> is the regularized incomplete beta function.

[edit] Shapes

The beta density function (not to be confused with the Beta function) can take on different shapes depending on the values of the two parameters:

<math>\alpha < 1,\ \beta < 1</math> is U-shaped (red plot)
<math>\alpha < 1,\ \beta \geq 1</math> or <math>\alpha = 1,\ \beta > 1</math> is strictly decreasing (blue plot)
- <math>\alpha = 1,\ \beta > 2</math> is strictly convex
- <math>\alpha = 1,\ \beta = 2</math> is a straight line
- <math>\alpha = 1,\ 1 < \beta < 2</math> is strictly concave
<math>\alpha = 1,\ \beta = 1</math> is the uniform distribution
<math>\alpha = 1,\ \beta < 1</math> or <math>\alpha > 1,\ \beta \leq 1</math> is strictly increasing (green plot)
- <math>\alpha > 2,\ \beta = 1</math> is strictly convex
- <math>\alpha = 2,\ \beta = 1</math> is a straight line
- <math>1 < \alpha < 2,\ \beta = 1</math> is strictly concave
<math>\alpha > 1,\ \beta > 1</math> is unimodal (purple & black plots)

Moreover, if <math>\alpha = \beta</math> then the density function is symmetric about 1/2 (red & purple plots).

[edit] Related distributions

The connection with the binomial distribution is mentioned below.
The B(1,1) distribution is identical to the standard uniform distribution.
If X and Y are independently distributed Γ(α, θ) and Γ(β, θ) respectively, then X / (X + Y) is distributed B(α,β).
If X and Y are independently distributed B(α,β) and F(2β,2α) (Snedecor's F distribution with 2β and 2α degrees of freedom), then Pr(X ≤ α/(α+xβ)) = Pr(Y > x) for all x>0.
The beta distribution is a special case of the Dirichlet distribution for only two parameters.
The Kumaraswamy distribution resembles the beta distribution.

[edit] Applications

B(i,j) with integer values of i and j is the distribution of the i-th highest of a sample of i+j-1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the i-th highest value is less than x, in other words, it is the probability that at least i of the random variables are less than x, a probability given by summing over the binomial distribution with its p parameter set to x. This shows the intimate connection between the beta distribution and the binomial distribution.

Beta distributions are used extensively in Bayesian statistics, since beta distributions provide a family of conjugate prior distributions for binomial distributions.

The Beta distribution can be used to model events which are constrained to take place within an interval defined by a minimum and maximum value. For this reason, the Beta distribution - along with the triangular distribution - is used extensively in PERT, CPM and other project management / control systems to describe the time to completion of a task.

[edit] External links

Beta Distribution, wolfram.com
Beta Distribution - Overview and Example, xycoon.com
Beta Distribution, brighton-webs.co.uk

Image:Bvn-small.png	Probability distributions [ view • talk • edit</span> ]
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